Gearing.



H n. WILLIAMS.

(BEARING. APPLICATION FHED JULY 25| IBIM RENEWED DEC- 5.191].

Patented Dec. 9,1919.

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H. D. WILLIAMS.

GEARING'.

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H. D. WILLIAMS.

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BEARING. APPLICATION FllED JULY 25. 1914. RENEWED DEC. 5. 1911.

Patented Dec. 9,1919.

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UNITED STATES PATENT OFFICE.

HARVEY D. WILLIAMS, 0}? WALLINGFORD, CONNECTICUT, ASSIGNOB, BY MESNE ASSIGNMENTS, T0 JAMES'E. GLEASON, TRUSTEE, OF ROCHESTER, N EW YORK.

GEARING.

Specification of Letters Patent.

Patented Dec. 9, 1919.

Application filed July 25, 1914, Serial No. 853,017. Renewed December 5, 1917. Serial No. 205,684.

To all whom it may concern:

Be it known that I, HARVEY D. \Vnmnnrs, a citizen of the United States, residing in the town of \Vallingford, in the county of New Haven and State of Connecticut, have invented'certain new and useful Improvements in Gearing, of which the following is a specification.

This invention relates to that class of toothed gearing in which the wheel member ofa pair of gears, has the teeth thereof angular y-disposed relatively to a line that is normal to the plane of revolution and is parallel to the axis of revolution of such wheel.

Thepresent appli'cationlis in part a continuation of my application filed April 14, 1913, Serial No. 760,861, on which was issued Letters Patent No. 1,112,509, dated October 6,1914, for improvements in bevel gearing. I

A principal object of the'invention is to furnish a system of toothed-gearing, in

' which a pair of the gears comprises a wheel having the working-surfaces of its teeth conforming to a sing]e-reproduction configuration, and a mating pinion having the Work-surfaces of its teeth conforming to a coacting compound-reproduction configw ration; thus the teeth of the gear-wheel and the teeth of the mating pinion therefor are of dissimilar formation, or configuration.

A further object is to furnish such gearing in which the profiles-pairs (through the length of the tooth-sm-faces of the wheel) are uniform in relation to a geometric master-form-axis and are also uniform in rclation to a: geometric plane in which said master-forn'i-axis is located, as hereinafter more fully explained.

A further object of my present invention is to furnish an improvcu toothed-gearing of that class in which the longitudinal line or geometrical axis of each pair of adjacent tooth-surfaces is disposed or located in an angular relation to the axis of revolution,

as in certain kinds of bevel and skew-bevel,

and in skew-spurgears; also to furnish such gearing in which the wheel, in the pair of gears shall constitute a master-wheel adapted for operating correctly not only with pinions of varying diameters but also with such pinions located with their axes in various degrees of divergence or axial skew, respectively, from an axial-plane radial to the master-wheel axis, and also relatively to the plane of rotation of the wheel. In this system of gearing, therefore, one skewtooth master-wheel may be practically and successfully employed in mesh with pinions of different diameters and having their axes of revolution located in different planes, respectively.

Another object is to furnish pairs of gears having superior operative features within a range of proportions which will include the larger classes of such gears, and thus utilize to the advantage of a large number of cases, certain features not equally applicable to the whole range of gear-wheel proportions. Accordingly, while my present improvements may not be suitable or well adapted for use in gear wheels of some proportions and sizes,-as regards, for instance, the pitch-circles and the relative numbers and sizes of the teeth,this restriction in the range of applicability may be considered as a means for securing desirable and important advantages in the mode of op-.

eration and in the manufacture of classes of gearing which are commercially of'much greater nnportance.

In my improved toothed-gearing, I pro- I vide the whecl,which will usually be the larger member of the pair,with working faces of a form adapted to be readily prccisioned, and I then arrange these working-surfaces in a longitudinal parallelism, and also in a transversely converging relation to each other. The generatedor evolution form of tooth-surfaces is thus restricted to one gear (the pinion) of pair, while the other gear, or Wheel, has? nonevolved teeth of a shape and relative size which are more readily vproduciblc with the required high degree of precision, and which are both producible and repairable by the use of simple and ordinary appliances and methods, and without requiring the use of any generating machines for forming or shaping the tooth-surfaces.

It will be remembered, that generally the rate of wear of the teeth on the two gears in any pair, is substantially'in inverse proportion to the numbers of teeth thereon. L'hus in a pair of gears having respectively twenty and sixty teeth, the wear of the teeth on the twenty-tooth pinion evidently would normally be three times as great as on the sixty-tooth wheel. The more rapid machines.

wear of the teeth of the smaller wheel, tends to continuously maintain and shape these teeth to the proper conjugate form for working correctly with the teeth of the larger wheel, provided. these wheel teeth are ot a precisionized character. Assuming that the gear'xand pinion ratios average three to one, then it is evident that threefourths of the Whole number of teeth, in this kind of gearing, can now be made without the use of generating machines, and without employing the evolution method, and still have the gears operate as truly and accurately as if the teeth of both gears were made lay that method in gear-tooth. generating In view of the intricate kinematical relations involved .in the art of toothed-gearing,--and especially when the teeth are run gularly-disposed, as in the present improvements,-I have herein employed the term wheel for designating the gear having the teeththereof provided with the working- ?faces having a single-reproduction configuration, and have designated the mating gear as the pinion, without regard, how- 6Y1','l30 their relati-i e actual sizes but having in mind that usually the wheel is larger than the pinion, and that while either one may be used. as the driver, the smaller said gear Wlll preferably and usually be the pun .1011 and be employed the driving member of thepair of gears. -In the accompanying drawings forming apart of this specification, my present in provements are generally illustrated iii-the several figures by a delineation of a frag mentary portion only of a complete wheel or pinion, as the case may be, since in any case the complete periphery or toothed-zone of the wheel is only a duplication of the construction within. one pitch-arc thereof; and similarly as to the pinion.

In the drawings, Figure 1 is a diagran1- matic View illustrating (but' in a general wayonly) certain relations of the wheel and pinion, and especially the relations of their respective tooth-surfaces with each other. e

Fig. 2 is a fragmentary sectional view. (drawn in almement with Figs. 1 and in side view, and which correspond in construction with those indicated in end View inFig. 1. I v

Fig. .3 is a similar fragmentary sectional view showing portions of a bevel wheel and .ion having teeth which are here shown in side View, and which correspond in com structien with those indicated in end View in Fig. l. i

Fig. n is fragmentary view drawn helow and in alinement with. Fig. 2, and showing in a portion of the spur-wheel B (Fig. 2.), and in plan or face view, a pair of tootlrsurtaces, as 7, 7?, arranged with a tooth-space between them and also located with an obliquity indicated by the angle e.

Fig. 3 is a similar fragmentary plan or face view of a portion of the bevel wheel B, (Fig. 3), showing a pair of the tooth surfaces, as f, F, arranged on the face or tooth-zone N, with the master-form axis :0 located in a plane of the axis-of-revolution, 0, of the wheel. i

Fig. 4c is a view similar to Fig. 3, but showing the pair of tooth-surfaces arranged on the tooth-zone or face, N, with an obliquity as indicated by the angle 6".

Figs. 5 and 6 are views in the nature of diagrams, and are arrangedin a perspective manner for illustrating certain features as regards the relative locations of wheel and pinion axes and tooth-surface obliquity, in the case of spur gearing in. Fig. 5 and of .bevel earinfl in Fi 6.

Fig. 7 is a diagram drawn in a manner similar to a perspective view, for illustrating the geometrical relations of certain features of the wheel member of a pair of the ears, and for showing the master-form axis in three arrangements or locations coinciding, respectively, with three locations of a certain hypotenuse line which is herc shown in the successive positions 00, a and a). 4 T

Figs. 8, 8 and 8 are fragmentary views (similar or corresponding to Figs. 1, 2 and 2", respectively,) drawn in projection with each other and illustrating pairs of tooth surfaces as seen in end-view, (Fig. 8), in side-view, (Fig. 8), and in plan or faceview, (Fig. 8), respectively, when the mas tor-form axis, x, is located in position 00 Fig. 7, and the wheel B is a skew-spur gear.

Figs. 9, 9 and 9 are fraglnentury views (similar or corresponding to Figs. 1, 3 and 3, respectively), drawn in projection with each other and illustrating pairs of toothsurfaces as seen in end-view (Fig. 9),in sideview (Fig. 9), and. in plan or face-view (Fig. 9") respectively, when the masterforin axis is located in position :10, Fig. 7, and the wheel, B, is a bevel gear.

Figs. 10, .10 and 10 are fragmentary views (similar or corresponding to Figs. 1, 3 and l, respectively) drawn in projection with each other and illustrating pairs of tooth-surfaces as seen in end-view (Fi 10), in side-view (Fig. 10),'and in plan or facc-view' (Fig. 10*), respectively, when the master-form axis, x, located in an inter-- mediate position, as as, Fig. 7, and the wheel B" is a skew-bevel gear.

Fig. 11 is a diagrammatic view showing a pair of wheel tooth-surfaces as seen in view (as in Figs. 1, 8, and 10) when the profile-lines,-as f f ,are arranged to coincide with a master-form that is symmetrically located with reference to a meridian line, as m passing through the master-form axis :10, and preferably located radially of the axis of revolution, (as 512 Figs. 1, 2 and 7) of the wheel.

Figs. 12 and 13 are views similar to Fig. 11 for illustrating certain features of the 1naster-form and of the arrangement of the profile-lines of a pair of wheel toothsurfaces.

Fig. 14 is a diagram for illustrating certain features hereinafter more fully explained, regarding the outward and the in ward arrangements of the convergence of the tooth-surfaces.

Figs. 1 5, 16 and 17 are diagrammatic views illustrative of the preferred method of making the gears when the wheel B (in a conical form thereof) has the master-form planes inwardly-converging.

Figs. 18 and 19 are views similar to Figs. 15 and 17, respectively, for similarly illustrating the method when applied to the making of gears in which the said master-form planes are outwardly-converging.

Fig. 20 is a side View, chiefly diagram matic, of-the pinion, P, as seen from the left-hand in Fig. 19; these two views are drawn in alinement to facilitate the comparison thereof.

- Fig. 21 is a face view of a portion of the pinion as seen from above in Fig. 20, with this difference however, that the tooth-zone is here shown as if thepitch-surface, (the geometric cone of revolution formed by themaster-form axis, when this is revolved about the pinion-axis), were unrolled into a plane, for thereby bringing all the pinionteeth to stand as if verticallydisposed in a plan view, to facilitate description and comparison.

Fig. 22 is a View inall respects similar to Fig. 21, with the exception, that whereas-in Fig. 21 the geometric master-form is applied to the tooth-spaces, (in correspondence with Fig. 19), this master-form is here shown applied in Fig. 22,'to the pinionteeth, thus producing the form of piniontooth having the longitudinally-parallel construction or configuration and with the intervening to0th-spaees of the longitudinally-tapering formation, in accordance with Fig. 17.

Fig. 21 is an enlarged fragmentary view showing a pair of pinion teeth correspond ing to those indicated in Fig. 21, as seen in the direction of the arrow in Fig. 20.

Fig. 22 is a view similar to Fig. 21, but showing pinion teeth and spaces arranged as in Fig. 22, and with these teeth having their wheel-engaging surfaces conforming to the compound-reproduction configuration, which is hereinafter more fully explained.

. Figs. 23, 24 and 25 are views similar to Figs. 15, 18 and 16, respectively, for illustrating the counter-part tools J and T for making pairs of tooth-surfaces in skew-spur wheels by the method of single-reproduction, and when the master-form-axis w is in the positions, Fig. 7.

Fig. 26 is an oblique view for more fully and clearly exhibiting the features indicated in Figs. 23, 2a and 25.

Fig. 27 is a view similar to Fig. 17, for illustrating the manner of applying the master-form tool T (Fig. 23) for making pairs of generated tooth-surfaces on the skew-spur pinion, by the method of compouml-reproduction, and for mating with a skew-spur wheel such as indicated iii-Figs. 1,

2, 2, in Figs. 8, 8, 8 25, 26.

Fig. 27" is a view drawn in alinement with Fig. 27, showing the skew-spur pinion, (as seen from the left-hand in Fig. 27), and Fig. 27 is a View of the said pinion as seen from above in Fig. 27.

Figs. 28 to 34 inclusive, are a series of diagrams for illustating,more especially,the geometrical relations, (at successive planes of revolution along the length of a pair of spur-wheel tooth-surfaces) of and in Figs. 23,

.the master-form axis and the trace-line thereof which constitutes an axial line within the mating pinion tooth, in a case where said master-form axis has a position or arrangement relatively to the axis of revolution, (00, Figs. 1 and 2) such as indicated by the hypotenuse-line w, in position m, Fig. 7

Figs. 35 and 36 are diagrams illustrative of certain features hereinafter more fully.

with and relatively to the pinion, a rolling movement in accordance with the geometric atch-surfaces. Thus the wheel member of a pair of the gears is a master-wheel which is operable with a plurality of pinions of different sizes, and located in various axial and angular relations to the wheel axis. Therefore, one such master-wheel may mesh with and drive at the same time several such pinions each having a different diameter.

the master-Wheel;

Or, one such master-wheel may similarly mesh with several pinions oi the same size but each having a diil'erent axial position relatively to the Wheel-axis. Of these two arrangements, the former is briefly and. dia grammatically indicated in Fig. 86, where the one master-wheel, l3, coacts at the same time with thepinions P, P and P each of a different size. The second said arrangenient is similarly indicated in Fig. where the pinions I, P and P are of the same diameter, but have each a dili'erent axial position relatively to the axis of a coact' ing mastcowheel, as B,---which may be the same master-wheel as designated by B in Figs. 2, 8 and 10. Any convenient p1urahty of such mating pinions may he oper" ated in meslrwith one and the same masterwheel, provided, or course, that each of said pinions so employed shall. be located free of the other pinions of any such assemblage or set; This peculiar and comparative unireu sality which I thus obtain as regards the pluralities of pinions,and as regards the varying arrangements thereof in which all of the pinions are correctly operable with one and the same mating wheel,hecomes greater in its range of practical application, in proportion to the increase of diameter of In a'pair of interxneshing gears, and as between the body of a Wheel-tooth and the body of a pinion-tooth, the proper rolling movement requires, in. any given instance, some aggregate amount of transversecurvature, or profile deviation, of the one tooth surface relatively to the other, and. in the former practice it was customary to appor tlion that total relative curvature one part to the Wheel and the remaining part to the pinion and to make a different a portionmcnt for each pair of gears having different relative diameters. In this system of gearing, and contrary to that former practice, no such separate apportionments are required, but the whole oi? such relative deviation may 'be applied to the pinion-tooth-surlaces This feature combined with the longitudinal-parallelism of the wire of wheel toothsurfaces, and with these surfaces transversely converging, is a means for bringing the two said members of the pair o't g into such a relationship that ad r having the improved operational features herein set forth, the wl1eel-teeth can ho'made on a series of wheels by the single-reproduction method and with the same mas tr-form and the same counterpart tool only the pinions require the compom r-reprod tion method for their manuiacturel improvement in the method oil manufacl 1, however, is not specifically claimed in, this present application.

In apair of mating gears 't' in only one direction, each of meteor teeth need. have as Well lrnown,ouly one wort.hag-surface, but these working-sun faces, in the gearing herein described, are of the single reproduction arrangement and ronligin-ation on the wheel-teeth and have uniform profiles throughout their length, while the coacting tooth-surfaces on the pinion teeth are conjugate to said wheel-tooth workingsurfaces and are of the compound reproduction arrangement and configuration. Eachwheel-tooth workingsurfacc has uniform profiles throughout the length there-- of and is of the single-reprodnotion configuration, but when each of said teeth has two such working surfaces, and. has these arranged in such relation that a pair of them (either two surfaces on one tooth. or two surwheel-tooth pairs, and which also, when.

taken as, or in pairs, conform to the com pound-ra nedaction arrangement and con figuration. In these arrangements, the

wheel-tooth working-surfaces are, oi. course,

transversely-cou verging, either outwardly, as inv Fig. 14-, or inwardly, as in Fig. ll.

"It will he remembered that the custom of making a tooth section symmetrical with a central line, as m Fig. 1i, (this line being located in plane radial to the axis of rota tion), is for the pro-pose of making the pair of gears operate the s cure in either direction. en the wheel are running in some ction (as, for instance, in the direction of the arrow '7', Fig. l) and the pinion i" the driver, only the rearward faces, as are the workingdfaces of the driven wheelteeth, while the opposite faces, as y, then operate as reaction faces for preventing an irregularity of movement because of any tendency to throw the wheels forwardly too rapidly, and thereby; regulate the reltive movements of the Wheel and pinion. with precision. Thus the act tion of the wheel B by the pinion I only 1.11 v ol res the said one curiae series of driven laces,- --or too of the wheel B (which. l dcei tion-faccria) and the coactr wMilly-acting or driving :lr---.- su'" 1c of the pin l.

.ally the larger gear, as E, of a pair of these oblique a regarded as being the who.

her is provided wi teeth which. have their surfaces arranged in transversely-converging pairs in which. each of said tooth-surfaces has a uniform profile throughout the length thereof. These tooth-surfaces are also arranged in pairs which are longitudinally parallel the one to the other, so that in any such pair of tooth-surfaces their normal operation involves a peculiar progressively meshing-action as between the wheel tooth-faces and the faces of a co-meshing pinion. This peculiar meshin action,-for the sake of brevity and for 518 want of a better term, I have designated as a skewaction. In this gearing,-therefore, there is a certain skew-action of the tooth-faces which is normal to the operation of the gears, and which involves a relatively progressive meshing, longitudinally of the teeth during the approach into full-mesh,-so that some variation in the skew angles does not,

in efiect, create a different kind of coaction,

although varying the same in both a qualitative and a quantitative manner. Thus a skew-action involving a longitudinally-progressive meshing, as between the workingsurfaces of the obliquely-disposed wheelteeth and the coacting but differently shaped surfaces of the pinion teeth, is obtained with piuions of different sizes and having respectively, different skew or angular locations of the pinion-axis relatively to the wheel-axis.

In the style or outline of the master-form F as shown in Fig. 12, this conforming to Figs. to '27, showing profile lines each of a uniform contour throughout the length thereof, as elsewhere hereinsct forth, the reversely-disposed profile-lines f and f are drawn as conic-section lines, since these pro file-lines are not only curred but are also circular in character; and these circular-arc, conic-section profile-lines are not only converging in relative arrangement, but are shown so proportioned that they meet on a line, as m. that is tangential to said profiles, and that constitutes a meridiandine for, or of. the master-form. In a similar manner, arcsof the other conic-section lines or curves may bc arranged as the profilelines of a master-form suitable for use in shaping by the methods of single-reproduction and coinpound-reproduction, respectively. the workin -surfaces of the teeth of the wheels and pinions, respectively, in pairs of gears made according to my present invention.

By the term conic, or conic-section as herein employed with reference to the pre f crable lines of curvature or uniform contour for the master-form profilc-lines, I have con sidered the circle as being comprised in the class of curves usually designated as conicwctions." and which are sometimes referred to bv the simple term conics.

fllhe aforesaid directly-coacting faces being arranged in converging pairs, and these 1 pairs, as i Fig. 14, and f and f Fig. 14, being alike except as to the direction of the convergence (here shown by arrows),

each such pair of the converging faces constitutes a structural feature which I designate as the master-form. A further feature of this master-form is the longitudinal parallelism of the said two transversely converging tooth-surfaces which are com- In a pair of these gears, when the wheel has the master-form with its center-line, as

0:, (see Figs. 5 and 9"), in the position on, Fig. 7 where said center-line is shown located in a plane radial to the wheel axis, the two tooth-surfaces, as f and f (see Fig. 3 have a skew-angle, a, relation to said wheelaxis, one of them, as f, has a rearward skew. Similarly, the pinion P has its tooth-surfaces arranged with their bounding master-form surfaces, and hence their surface elements in general,-located on similar skew-angles, and these in relatively the same order of arrangement. When the action-face, as f of the wheel has-a rearward skew. (see Fi 3") the' amountof which is indicated by the angle a, the coacting pinion tooth-surface will have a co responding amount of skew in the'said direction. In this pinion construction, therefore, the master-form of or for the pinion tooth-surfaces has the two sides thereof in longitudinal parallelism, in this respect corresponding with the wheel construction. When the wheel master-form has the 'inwardly-converging longitudinally parallel surfaces (and therefore has the parallel spaces, as F) the pinion masterform has,relatively to the pinion pitchsurface,the outwardly-convergin g arrangement of the said master-form boundingsurfaces or tooth faces. Therefore these two master-forms come into the samepositions and coincide when the two engaging toothsections also come to the exact full-mesh position, this being illustrated in Figs. 8, 8 and i0, 10.

In each of the described arrangements of the convergence of the said wheel tooth-surfaces,as indicated for instance in Figs. 8, 8 and 10, 10*, respectively, these wheeltooth surfaces. (in any series of the masterforms having the. longitudinally parallel contruction), are so arranged and related and successive points along this line (w succeeding to the other of them in the circumferential tootlrzone N of the wheel.

' This'peculiar relationship and mode of co- 15 action, is indicated in Figs. 3" and l: and other views, and particularly in said F g. 3, where the two inwardly-converging successive faces 21 and 22 are in parallel, while the next successive face, at an angle, as Z, thereto. Also in said Fig. 3, another set of three successive faces comprises the two outwardly-converging longitudinal non parallel faces 22, 23, and the next succeeding pa 'allelly-disposcd face 2 1-. In each of these two sets or groups, the directl opposite action and reaction is indicated by the parallelism of the two oppositely-acting faces, as 21, 22, in the one set, and 23, 2 1 in the other set.

The sheet of drawing crnn iirising Figs. :28 to 34", inclusive, constitutes a single and comprehensive diagram, which composed of several series of small diagrz'uns that are separately designated, and are drawn in alinement, or projection, in a conventional manner, to facilitate comparison. In the plan-view diagram Fig. '29, the line :0 is the same master-form axis, 0?, shown in F the length of a tooth-space) are shown at a", in Figs. 30, 31, 32, and 34, respectively; also, the line no in Fig. 25% is the line of Fig. 29, as it would appear when somewhat produced and then seen from below in Fig. 29. In these views, the profiles and the point (1)., correspond in their relations, with the same elements as illustrated in other figures, including F igs. 11, M and For convenience, only live of the small diagrams are shown in one series; Fig. shows the master-form and profiles-position as in a mid'plane of rotation, inthe length of gear tooth-space, when the gear is rotated to bring the line a to coincide with the geometric instant axis; Figs. 30 and 34, represent those relations in planes of rotation at the ends of the tooth-space, while Figs. 31. i 3, represent the said relations at intermediate planes of rotation along the axis. 13 of the wheel. The line 0/), Figs. 28, being located ir. a diagonal position and also a straight line, (corresponding to line 01 Figs. 7, 2 only one point therein comes to the instant axis in the geometric rolling surfaces of the wheel and pinion, and this deviation is shown by means? the divergent positions of the lines as and V inl igs. 28, 529, also in the views Elm-34, and 30 -s-3 ll.

As regards the relations of the coacting pairs of wheel'tooth working-surfaces and the corresponding pinion-tooth wbrking sur faces, these progressive relations are further exhibited in Figs. 28 to 34, inclusive, and they may be said to be derived from and to follow the master-forni-axis a: (see Fig. 29) as regards the wheel-tooth surfaces and the corresponding trace-line V, as to the piniontooth surfaces. These relations, therefore, involve the single reproduction and the con figiu'ation resulting therefrom, as compared with the compound-reproduction and the peculiar curved surfaces resulting therefrom But as regards the relation of the profiles oi the said pairs of surfaces to a radius of the wheel at any particular point in the length oi the tooth surface, (that is with relation to anv )lane of rotation these rofiles are of course, determined by the single reproduction method as regards the wheel-to0th Working-surl aces (as in Figs. 30 to 34), while the coacting and conjugate pinion-tooth prolilcs have a curvature which results from and corresponds with the epicycloidal relations at successive points along the instant axis, as m, Fig. 7, or m, Fig. 2*. Thus, it

. may be said that the general proportion and torn'urtion of the said two pairs of surfaces oi the wheel and pinion, respectively, are the result of the single reproduction and the compound reproduction (as elsewhere herein more fully described) with relation to a master form axis which, as regards the wheel, lies obliquely to and outside of the pitch surface; and also with relation to a pinion-tooth axis whichv lies within the pitch surface of the pinion. Successive positions ol. the tracedin-e V are shown in side view in Figs. 30 to 34. while the same positions are shown in end View, in Figs. 30" to 34. re-

spcctivelv; both of these series of views are drawn in projection with the plan-view diagram, Fig. 29, as shown by the usual dashlines. In all these views the normal,.or more usual, deviation of the trace-line V relatively to, or away from the axial line as, is somewhat exaggerated so that the diagrams may be more easily compared. Fig. 28 corresponds in position with Fig. 32".

, Referring to the diagram F ig.'7 if a line, as 2: which is a normal to the axis m at the point (3, be rotated about said axis, the outer end or said line 2 describes the meridian circle m, located in a plane-of rotation. At some other point, as c, on said axis 12 describe a similar circle as m also in a plane of rotation; the two circles then lie in parallel planes and these planes are at rightanglcs to the axial line 0: which represents the axis oil-rotation of any gear or Wheel that shall be drawn out or designed in-accordance' with the diagram thus arranged or proportioned. In practice, as will be evident, the diagram, Fig. 7, may be made of such proportions are suitable for meeting the requirements in any given pair of gears to be made in accordance with my present improvements. it will be seen that the circle m is an orbital path in which the point 2 may be said to revolve around the point 9 of the circle m Said orbital circle m is shown divided by the lines 3 3 into arcs of which the quadrant 0 to z is the lower'righthand quadrant, extending in the circle m from the line as, at the point 0, to the line a, at the point z,these lines a: and 02 being diiierent positions of a hypotenuse line.

The said hypotenuse line, as 50 at the vertex am: of the triangle 50w, 3 a, Fig. 7, inteisects or contacts with the instant axis m, and this is in a plane of rotation which, for convenience, I designate as the meridian plane, and in which lies the circle m this plane is also the profile meridian. W hen the described locating-triangle was, 3 .2, Fig. 7, has the plane thereof, (which is indicated by the line y 2,) located in a tangential relation to the meridian-circle.m",and also tangential to the geometric surtace-ot-revolution generated by the maste1'-foi'm-axis revolving aboutthe wheel-axis 0 and the tooth-zone of the wheel, as N, Fig. 2, being located contiguous to said nieridiaireircle m the geanwheel (see Fig. 26) is then of the particular kind which 1 designate as skew-spur gearing.

In a pair of these gears, when the wheel has the master-form with its center-line, as m, Fig. 7 in the position :0, (which is also in a, plane radial to the Wheel axis) the two tooth-surfaces, as f and f (see Fi a 3) have a skew-angle relation to said wheelaxis, one of them, as 7, having a forwa'd skew while the other said face, as P, has relatively a rearward skew. Similarly, the pinion P has its tooth-surfaces arranged with their bounding master-form surfaces,- and hence their surface elen'ients in general,-located on similar skew-angles, and these in relatively the same order of arrangement. \Vhen the action-face, as f, of theuvheel has a rearward skew, the amount of which is indicated by the angle a, the coacting pinion tooth-surface will have a corresponding amount of skew in the said direction. In this pinion construction, therefore, the master-form of the pinion tooth-surfaces has the two sides thereof in longitudinal parallelism, in this respect corresponding with the wheel construction.

But when the wheel IlltlStQR-fOllfl has theiuwardly-converging longitudinally paral lel surfaces, (and therefore has the parallel spaces. as F (Fig. 1) the pinion mastertorm has,--relatively to the pinion pitchsurface,the outwardly-converging arraiigement of the said master-form boundingsurfaces 01' tooth-faces. Therefore these two master-forms come into the same positions and coincide when the two engaging toothsections also come to the exact full-mesh position, this being illustrated at :12, Fig. 1.

From the foregoing description as herein illustrated, it will now be evident how the master-form-axis is not only a line located in a plane, but also is a line having an angular position relatively to the wheel-axis, to the plane of rotation and to any straightline that is a normal to (vertical to) such plane of rotation; and how, in the pinion, the corresponding tooth-surface axial-lines (as V, Figs. 28, 2.9) ,of the respective toothsurface pairs,-are rotational trace-lines of said wheel master-form-axes, m, as more. fully explained in connection with said composite diagram, Figs. 28 to 34".

In practice, any desired number of exchangeable pinions of different sizes respectively, may have their working-surfaces shaped by the method of compound-reprbduction, or evolution,. fron'rand 'by the counter-part of the same master-form F, by generating the pinion teeth from a rolling movement on the geometric itch-surfaces. In this operation pitch-surfaces resulting from the rolling movement will, of course, meet on an axial line, as in, Figs. l7 and 27. In any such case, however, the same counterpart tool, as T, having a profile outline oo- .inciding with'said master-form (or a suitable correspondingly-shaped millingcutter,-not shown,in lieu thereof) Will properly geiierate the corresponding pinion-tooth eurved-surface of the conjugate form for working correctly upon the tooth-surfacesof such nrasteuwheel. ihus a single masterwhcel may constitute a master or form for generating the teeth on any plurality of pinions of different sizes, respectively, and each specially shaped for rolling correctly in mesh with that one master wheel; but these diiierent pinions, being each thus specially shaped, are not thereby "formed for working in mesh with each other, nor with a master-wheel of a diii'erent size or proportion. In all such instances, however, the pinion tooth-surface may be properly described as having a developed curvature generated according to the tooth-surfaces of the mating master-wheel of the pair. .These pinion tooth-surfaces are said to be generated because in practice they will be, usually and preferably, produced in and by some suitable gear-tooth-gencrating machine, the tooth-surfaces being developed from and by a tool whose cutting lines are made according to the form of the master-wheel pairs oftoothsurfaces, so that the resultant or conjugate tooth-face transverse curvature will be of the same developed formation as if the tootlrsurt'eccs of the ohn ion Were actually molded. into shape hy the rolling of the pinion in with its own .master-Wheel. 5 As on eid'to efnll. end clear description of the veering, I will. now hrietly or ii one mother. of melting thenmsteniorin out oi? the Wheel, (first referring to conical wheels having obliquely-{lisposed tooth-sin.- iscee) end the use of the counterpart tools for melting of the coniugntionelly-curred pinion-tooth surfaces by the duplicate of the ninsteriorin of the Wheel with which such pinion is to In l llg. 1:") the ellipse 8 is the, outer circle of the Wheel B, ns seen the d: 7 sl'lown in Fig. 10. the lineot parallel to the axis ot the re. outed master form F; this view shows the Worlririg-curiae s of said masterforin arranged inwerdly-converging In .llig. 16 2t section is shown of the tooth-rim of the Wheel B, as seen from the right hzind in in: 1-5, the tooth-surface i being shown formed, While the muster-termaxis e also. the line of movement of the file-outline oil? which e. 15) ond with st l lr heiug'g prope nu'ner inn suitehl I 5; man-lune (not sliiown), and ll being); rcciprocuted While Carried in tlie usual tooth-holder of such wellslrnown shep- .uiechine the tool ronytlicn he ted rradunlly downvmrd to its seid position in l l5 and 16, ther Jy reproducing tee term oi? the tool in :1 reversed urrnugenlent in the wheel rim and with only it single direction or kind of movement es between the "heel. and the to l and thus iuplet' purull l loth-surteci my the metlwd. l 'his opernt posted around the Wheel at each interval, all. the tooth-feces will he 0" ill ig. 15.. tor the purpose of commir of tools n. counter-poi ool, llnl'iCll by dotted lines in :i position in which it exactly matches the aforesaid. tool .I. and is on exact counterpart of the tooth 'lori'n F of the Wheel. Such a counterpart tooLis shown at T. Fig. 17, as employed for mnlnu "he pinion-tooth g of e conjugate curvature for correctly operating "with the aforesaid tooth-form ili'uces and f of the wheel. lor this purpose. the tool T may he liven the Well-known planer increment" in the line oi the tooth-form enis r (coinciding with the line oi sight in Fi s. 1 and i l) and at the same time will swing in n circle. as 8 (horn-in seen 21.; an clliy coinciding with the ,s'wnuctric pitch-s1 race of the Wheel; that is. the tool "ll has a. compourul inc tiou, comprising u Lcciprocet -y or nothing Ell movement in the line of. the instantexis oi? the Wheel, While this line of movement also revolves about the axis of the Wheel. It Will be seen, however, When the Wheeltooth profile is e straight-line, such a. coincidence of the line of tool movement with the intent axis is not required. During this compound or generating operation, the pinion blank: P also revolves, (as'indicated by arrows in Fig. 17) until the tool T has swung from the initial position T to the final position T, thereby completing both of the feces oi the piniou-tooth g by a single 19))9111171011 of the machine, and by n method of conipmind-repreduction whereby the copuplete master-form with its surfaces of uniform profile and parallel construction is reproduced in n conjugation el manner in the or ed feces of the similarly parallel pinion teeth such as shown ohliquely'loeated in the pinion body]? in Fig. 22.

Figs. 18 end 19 correspond to said Figs. and 1'1, respectively with the exception that the tools J and T have changed places, the tool T being shown in Fig. 18 as used for melting the heel-tooth, and with the master-form faces outwerdly-con'verging, but with the single-reprod'uction as before. In the connienion View, Fig. 19, the counterpart tool J is shown as used for generati the corresponding and oppositely-disposed curred feces of the pinion-teeth, with and between which the said Wheel-tooth g is to directly vco-zict. Thus in each arrangement of the master-form, the two counterpert tools by one operation of each of the tools produce four tacos arranged in two the lines of parallel 1'1121St1fOI' nS which are identical tor both the Wheel and the pinion.

In Fig". 20. the pinion P shown in correspondence "with the same member in Fig. 1'9, as to the lllClll'lBCl position of the pinion axis, it. which is here shown inclined for hringinp; the instant axis, at, into elineinont With the point a:, in Fig. 19,--this point l 'ir n Figs. 17 end 19, the projection of :e.nt-nrris. In Figs. 21 to 2%, the red lines '2 /1, indicate the surface-elements "which. come to parallelism when the mating. I

wheel and pinion teeth come to e symmn iicel position, one instance of Which illustrated in Fig. 9, for the form of tooth shown in Fig. This is perticulrly indicated in Fig. 9.1 where the parallel lines '2". n the hearing lines of which the lines 2', 1?, (l ig'; 21.) are projections. The curved character of the working surfaces of the tooth 5 21, is indicated by the curveture of the lines already described, and is further imlicnted in l ig's. E21, 21, by the slightly curved bottom and top lines and j. rcs uu-tiroly, but those curvatures are slightly exaggerated for melting these die:

einlnetic representations appear more clearly on the drawing. For the tapering form of pinion-tooth, as in Fi s. 21, 21*, a corresponding symmetrical position is indicated in Fig. 19, where the wheel-tooth po- .lines. This Fig. 20, although drawn in projection with Fig. 19, would also be correct if similarly drawn in projection with Fig.

17, since, on the scale here illustrated, such.-

a side view of each form of pinion-tooth will be nearly the same. In Figs. 21 and2l the pinion-teeth g are shown having the longitudinally-parallel configuration, in accordance with Figs. 10 to 10" and when F is assumed to be awheel tooth-space and h is a tooth; but in Figs. 21 and 21, the pinionteeth, g, have the described tapering formation, and also the obliquely-disposed loca-- tion.

The form or configuration of a tooth- ;surface may be said to include the features of outline, or profile, and the relative position thereof. The character of the profile depends, of course, on the kind ofreproduction, whether single, (as in the case of the wheel),-or compond, as in the case of the pinion; the position depending on the direction of the transverse convergence,

whether inwardly or outwardly. As thus applied to the wheel, each profile or sideline, as f or f (Figs. 12 to 14) of the form F, gives,always by the same direct re- 'production,the shape or profile, and also the position or angular relation, to one tooth-surface in each said arrangement of the wheel construction. In the primary arrangement, the said master-form,-as represented in a tool (as T, Figs. 17, 27 that is coincident therewith,and by the use of the single-reproduction method, produces a pair of tooth-surfaces bounding the body of one and the same tooth, (Fig. 14), and therefore gives to this tooth an actual sectional shape and size which is the exact counterpart of said geometric master-form. In the secondary arrangement, the same results as to outline and relative position are produced (as by the tooth J, Fig. 19), on the two adjacent tooth-surfaces of two adjacent toothbodies, respectively, but with the sectional shape and size applied to the tooth-spaces.

In practice, it is desirable that the angular variations of the gear-axes shall only be adopted within moderate limits, but these limits, however, may vary somewhat from the specific proportions herein illustrated. The limits within which the relative location of the pinion axes may be varied as herein described, (as in Fig. 35, for instance) can readily be ascertained in practice by trial or by the aid of suitable graphic methods, having in view the extent to which in any given instance the relative obliquity which may be desirable as between the pinion tooth-faces and the pinion-axis; in most cases an angle of obliquity of about 10 degrees is deemed to be within practicable working, limits. The ran 'e of variation which in practice it may be feasible to adopt,

as to axial obliquity, I have herein designated as the cooperative relation of the pinion axis relatively to the wheel axis. In this connection it will, of course, be understood that the specific character or conjugate shape of the working surfaces of the pinion teeth will vary, in pairs of gears otherwise the same but having some-particular axial obliquity in order to make-such pinion surfaces properly conform to the required compound reproduction arrangement and configuration. 4

Since each wheel-tooth working-surface has uniform profiles throughout the length thereof and is of the sin 'le-reproduction configuration, when each sald tooth has two such working surfaces, and these are arranged in such relation that a pair of them (either two surfaces on 'onetooth or two surfaces on each of two adjacent teeth), are formed in parallel, then such pair of work ing-surfaces is said to conform as pairs to the single-reproduction arrangement and configuration; and, -conversely, the coacting pinion-tooth working-surfaces will then be formed in pairs which are said to conform as pairs to the compound-reproduction arrangement and configuration. When thus organized, each pinion-tooth is provided with a pair of working-surfaces which are individually conjugate to the respective working-surfaces of said wheel-tooth pairs and which also, when taken as'or in pairs, conform as pairs to the compound-reproduction arrangement and configuration. The obliquely-located wheel-pair of said tooth surfaces being of an untwisted form or construction, it follows that the mating pair of pinion tooth-surfaces will be not only obliquely-located but also will have a twist throughout the length thereof; this is especially illustrated in the sheet of diagrams, Figs. 28 to 34L", in which the described twist is indicated in Fig. 29 by the divergent trace-line V relatively to the wheel master-- form axis as.

The two described arrangements of the transverse convergence of the wheel-tooth No. 1,112,509, and therefore I have only I briefly described this feature in the present application. Those two constructions as regards the transverse convergence are re garded as broadly included under one definition and hence are herein so l$lb$i and claimed: hut the inwardlyconverging construction regarded as having some important adva ges, and especially since this form of the ieel has,as compared with the othcr,--u versed angular, or circuml iva cc of the toot-l1-i'ace surall? driving {or Motion) of the wheel-teeth, and has sit erection and other special qualities and features of operation. as else- Where hereii fully set forth.

In the orgainzation in the mating Wheel and pinion, the respective working-surfaces into the angujlarlywlisposed coacting sprhice-pair --and in whichever direction the trans convergence may be arranged,it will be observed that each of the pinion-tooth working-Silviaces is conjugate to one series only of the wheel-tooth working-surfaces, that is to those facing in the oirection; and therefore, it will be t. u that the workingsnrfaces comprised in the two series o'l wheel-tooth working-surtaces, respectively, maybe of (liilerent profiles, and may have different pressure en; es. tint in any such instance the two pinion worldng-sinffaces which conform as a pair to the co:npound-reprodue tion configuration and arrangement, will be individually conjugate to th two wheel workingsurtaces, inept-actively, which conform as a pair to the single-reprodnction configuration and arm gement.

By reason of the sameness of organiza- 5' tion as forth of the mastet-itorans of both wheel a A relative diiectiou of the the ivluel ill tl pinion, re of a di oi thine master-forms, and 52th 24) when the court '27) is cor s iverg'ence (in use ihed u i. that this I ound opera- 7' at any one point. may '3 out image imtii th" ed t *l. .oinpletely, pod,

and with vthe sult oi iorniing the opposite tooth-surfaces of a transv selycurved 5o shape or; profile which has a. col'istantly chimp-leg" prolile-cnrvaturc throughout the lcn, h'o'ilthe. pinion tooth, and thereby prodncii Jon the pinion, a tooth chara rizcd [feature of having the oppositie workitt'ace }f)1llTlillY elements in para-lielism. 1e pinion l Figs. 2h, :22. the showed in g, are indicated as being of that form, in primary elements heing hidicated by he lines and 2 it havin a given iitch.-this system. as will.

c: w 1 n pinion and the raven-cl of the new be evident, has the further advantage of making the tooth-spaces of all sizes and varieties of pinions for such Wheels by the use of only-one form of tool, and this of only one size, operating by the method of 70- compound-reproduction, or generation, now sometimes designated as evolution. A direct result ofthose features is that a single air of the counter art tools suffice for making all the'pairs o the gBErs (regardles of 7.5 varying diameters and different obliquity angles), of a given pitch and size of tooth, thus reducing the cost of tools and of tool gages, and the cost of their maintenance, to the very lowest practicable basis. An immediate and important practical result of this system of parallelconstruction combined with angular-location as applied to both the pinion-teeth and, wheel tooth-space from identical tooth-form profiles (in addition to the great advantage and economy,as already explained, of eliminating the compound -reproduction otherwise reql'lired for making the Wheel teetln) is the complete elimination ofthe difficult setting and gaging operations which are necessary, as regards both Wheel and pinion, in the manufacture of gears having teeth of a cross-section which varies in form or size, or both, along the length of the tooth. Also in this parallel-com strnction, the pinion tooth may be completely shaped and sized without the danger of a variation in Widths of successive teeth being caused by any inaccuracy in the 0 indexing of the wheel blanks, when the "teeth are made by cutting operations; thus I. avoid one of the causes of imperfection appertaining to the manufacture 0; skewtoothed, or oblique, gearing in gear-toothgenerating machines. In the operation of the machine, in making the wheel, the reciprocating tool at all points in the tool-section,-includingv(of course} all points in the pr0file-l moves in lines parallel to the line wmcyi, (in the symmetrical position of a pinion tooth), lies in (or in touch with) the geometric pitch-surface, and t'rom which in the rolling movement of the ears, the simihuline oi the pinion describes an epicycloid at the point or points thereof which in said movement lie in both pitch-surfaces while at said symmetrical position. This is indicated in li'ig. 17 by the paths 20, 20, of the master-form axis of the pinion, this axis being. so to speak, merged at in with the corresponding wheel master-form axis, since in the symmetrical position shown in Fi 3.7, (also in Figs. 1,8, 9, 10, 19, 33, 24 and I25 27). those two master-form axes coincide, that is. occupy the same ositions.

From the diagram ig. 35,-especially when compared with Fig. 7 and other views as herein described,-it will be evident that when the Wheel-axis and the pinion axis are not located in parallel planes, the obliquity or angular position of the working-surfaces of the teeth of the pinion will be correspond- "said pinion axial positions;

.'-This invention, as will now be evident,

furnishes a means and system whereby angularly-disposed-or oblique-tooth gears operating correctly as skew-gears, are readily and cheaply producible Without resorting $0 the usual but difiieult forms involving the hyperbola-of-revolution. Furthermore, the

described skew-gears may now be made with the same economy and facility as non-oblique gears, and may be produced in and by the same machines and appliances. These improvements, therefore, practically eliminate from the gear-making art, one of its most difficult and unsatisfactory branches, and thus make available for a wide range of applications, the valuable features of the oblique-tooth gearing for use in many situations where, hitherto, the difficulty of manufacture and consequent high-cost have been prohibitive.

In my Patent No. 1,112,509, I have illustrated and described certain features of my present improvements as applied to toothedgears of the bevel and skew-bevel kinds, and which have their teeth located at an angle to the axis of revolution and have the Wheel-tooth working-surfaces made with a single-reproducti on configuration that is distinguished by a master-form of straight-line profiles and tooth-surfaces coinciding; with geometr idplanes; these particular features,

of true planes. This application, therefore includes under one definition, not only the bevel and skewbevel forms of a certain kind of gearing and as to certain features thereof, but also includes, under the same definition, a corresponding form of skew-tooth spur-gearing,either with or without an axial skew,-in which the wheel-teeth have working-surfaces conforming to the singlereproduction configuration, and in which the mating pinion has. coacting working-surfaces conjugate to said wheel-tooth surfaces and conforming to the compound-reproduc tion configuration.

In a copendiug application, Serial No. 325,216, I have described and claimed subject-matter which is disclosed in part in this present application, but which is not specifically claimed therein.

Having thus described my invention, I claim f I 1. The herein described improven'ient in toothed-gearing, it consisting in a pair of gears comprising a master-vvheel provided with teeth each having a working-surface with uniform profiles throughout the length thereof and each surface having surface-elemerits in alinement with a geometric masterform axis. which coincides with the hypotenuse of a plane-triangle another side of which, at a plurality of points in the length thereof, coincides with a line that is parallel to the wheel-axis, the vertex of said triangle being in the instant-axis of the gears,'and said wheel-tooth working-surfaces conforming to the single-reproduction configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel axis, and having wheel-engaging teeth each provided with a workingsurface conjugate to said wheel-tooth working-surfaces and conforming to the compound-reproduction configuration.

2. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-wheel provided with teeth each having a working-surface with profiles of uniform curvature throughout the length of the-profile and each surface having straight-line surface-elements in parallel with a geometric master-form-axis which coincides with the straightline hypotenuse of a plane-triangle another side of which at a plurality of points in the length thereof) coincides with a line that is parallel to the wheel-axis, the vertex of said triangle being in the instant-axis of the gears, and said wheel-tooth workin -surfaces conforming to the single-reproduction configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheelengaging teeth each provided witl. a working-surface conjugate to said wheel-tooth working-surfaces and conforming to the conipoulid-reproduction configuration.

3. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-Wheel provided with teeth each having a working-surface with profiles of uniform curvature throughout the length of the profile and each surface having surface-elements in alinement' with a geometric master-form-axis which coincides with the hypotenuse of a planetriangle another side of which at a pluduction configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheel-engaging teeth each provided with a Working-surface conjugate to said wheeltooth working-surfaces and conforming to the compound-reproduction configuration.

4. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprismg a master-wheel provided with teeth each having a pair of transversely converging working-surfaces of which each surface has straight-line surface-elements in parallel with a geometric master-form axis which coincides with the straight-line hypotenuse of a plane-triangle another side of which is parallel to the wheel-axis and the vertex of which is in the instant-axis of the gears, said wheel-tooth pairs of workingsurfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheel-engaging teeth provided with working-surfaces which are conjugate to said wheel-tooth working-surfaces and conform to the compound-reproduction arrangement and configuration.

5, The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-wheel provided with teeth each having a pair of transversely and inwardly converging Working-surfaces of which each surface has straight-line surface-elements in parallel with a geometric master-form axis which coincides with the straight-line hypotenuse of a plane-triangle another side of which is parallel to the wheel-axis and the vertex of which is in the instant-axis of the gears, said wheel-tooth pairs of working-surfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheel-engaging teeth provided with working-surfaces which are conjugate to said wheeltooth working-surfaces and conform to the compound-reproduction arrangen'lent and configuration. v

6. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-wheel provided with teeth each having a pair of transversely converging working-surfaces with uniform profiles throughout their length and each surface having surface-elements in alinement with a geometric master-form.-axis which coincides with the hypotenuse of a plane triangle another side of which at'a plurality of points in the length thereof, coincides with a line which is parallel to the wheel-axis, and the vertex of which is in the instant-axis of the gears, the plane of said triangle being oblique to and intersec ing a plane through the wheel-axis and said ver tex, and said wheel-tooth pairs of workingsurfaces conforming to the single-reproduction arrangement and configuratiom'in com bination with a mating pinion having its axis in cooperative relation to the wheelaxis and having wheel-engaging tooth each provided with a pair of working-surfaces which are individually conjugate to the respective working-surfaces of said wheeltooth surface-pairs, and which in pairs conform to the compoundreproduction arrange ment and configuration.

7. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-wheel provided with teeth each having a pair of transversely converging workingsurfaces with uniform profiles throughout their length and each surface having straight-line surface-elements .in parallel with a geometric master-form axis which coincides with the straight-line hypotenuse of a planetriangle another side of which is parallel to the wheel-axis and the vertex of which is in the instant-axis of the gears,the plane of said trianglev being oblique to and intersecting a plane through the wheel-axis and said vertex, and said wheel-tooth. pairs of working-surfaces conforming to the single-reyn'oduetion arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to the Wheel-axis, and having wheel-engaging teeth each provided with a pair of work1ng-surfaces which are individually conjugate to the respective working-surfaces of said wheel-tooth surface pairs, and which in pairs conform to the compound-reproduction arrangement and configuration.

8. The herein described improvement in toothed-gearing it consisting in a pair of gears comprising a master-wln-rel provided with teeth each having a pair of transversely and inwardly converging workingsurfaces with uniform profiles throughout their length and each surface having surface-elements in alinement with a geomctl ric master-form axis which coincides with the hypotenuse of a planetnangle another side of which at a plurality of points 111 the length thereof, coincides with a line which is parallel to the wheel-axis, and the vertex of which is in the instant-axis of the gears, the plane of said triangle being oblique to and intersecting a plane through the whcch axis and said vertex, and said wheel-tooth pairs of working-surfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheelengaging teeth each provided with a pair of workirig-surfaces which are individually conjugate to the respective workingnsur' which is in the instant-axis of the gears, the

plane of said triangle being oblique to and intersecting a plane through the wheel-axis and said vertex, and'said wheel-tooth pairs of working-surfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to the wheel-axis, and having wheel-engaging teeth each provided with a pair of worl; ing-surfaces which are individually conjugate to the respective working-surfaces of said wheel-tooth surface-pairs, and which in pairs conform to the compound-reproduction arrangement and configuration.

10. The herein described improvement in toothed-gearing, it consisting in a pair ofgears coinprislnga master-wheel provided with teeth each having a pair of transversely converging working-surfaces with profiles of conic-section contour and uniform throughout the length of the tooth surface, and each surface having surface-elements in alinement with a geometric master-form-airis which coincides with the hypotenuse of a'plane-triangle another side of which at a plurality of. points in the length thereof, coincides with a line that is parallel to the wheel-axis, the vertex of thetriangle being in the instant-axis of the gears and the plane of said triangle being oblique to and lntersecting a plane through the wheel-axis and said vertex, and said wheel-tooth pairs of working-surfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in c0- op-erative relation to the wheel-axis, and l aving wheel-engaging teeth provided with working-surfaces which are conjugate to said wheel-tooth working-surfaces and conform to the compound-reproduction ar .rangement and configuration.

11, The herein described improvement in toothed-gearing, -it consisting in a pair of gears con'iprising a master-wheel provided with teeth each having a pair of trans versely and inwardly converging working surfaces with curved profiles of conicsection contour and uniform throughout the length of the tooth-surface, and each surface having surface-elements in alinement with a geometric master-form-axis which coincides with the hypotenuse of a'planetriangle another side of which at a plurality of points in the length thereof, coincides with a line that is parallel to the wheel-axis, the'vertex of the triangle being in the instant-axis. of the gears and the plane of said triangle being oblique to and intersecting a plane through the wheel-axis and said vertex, and said wheel-tooth pairs of workingsurfaces conforming to the single-reproduction arrangement and configuration, in combination with a mating pinion having its axis in cooperative relation to. the wheel-axis, and having wheel-engaging teeth provided with working-surfaces which are conjugate to said wheel-tooth working-surfaces and conform to the compound-reproduction a1"- rangement and configuration.

1:2. The herein described improvement in toothed-gearing, it consisting in a pair of gears comprising a master-wheel provided with teeth each having a pair of transversely converging working-surfaces with profiles of a conic-section contour consisting of a circular-arc and uniform throughout the length of the tooth-surface, and each surface having surface-elen'ients in alinement with a geometric master-form-axis which coincides with the hypotenuse of a plane-triangle another side of which at a plurality of points in the length thereof, co-

incides with a line that is parallel to the,

wheel-axis, the vertex of the triangle being in the instant-axis. of the gears and the plane of said triangle being oblique to and intersecting a plane through the wheel-axis and said vertex, and said wheel-toothpairs of working-surfaces conforming to the single-reproduction arrangement and configuration, in combination with a matin pinion having its axis in cooperative re lation to the wheel-axis, and having wheel-engaging teeth provided with workingsurfaces which are conjugate to said wheel-tooth Working-surfaces and conform to the compound-reproduction arrangement and configuration.

1.3. The herein described improvement in gears comprising a master-wheel provided with teeth each having a pair of transversely and inwardly converging working-surfaces with profiles of a conic-section contour consisting of a circular-arc and uniform throughout the length of the tooths 'u1face, and each surface having straight-line surface-elements in parallel with a geometric master-form-axis which coincides with the straight-line hypotenuse of a plane-triangle another side of which coincides with a line that is parallel to the wheel aiis, the vortex toothed-gearing, it consisting in a pair of 

